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No. Consider an analogous question for the random variables $Z_1$, $Z_2$ uniformly distributed on $\lbrace0,1,2,3\rbrace$ whose sum has the probabilities $1/16, 2/16, 3/16, 4/16, 3/16, 2/16, 1/16$ of …

The probability that there is at least one bin with at least $c$ items is less than or equal to the expected number of bins with at least $c$ items, which is $k$ times the probability that a particula …

There are many more partitions than there are pairs for large enough $n$, so rank-nullity says uniqueness will fail. $\frac12 (12) + \frac12 (34) \sim \frac12 (12)(34) + \frac12 e$ where we identify …

The minimum value is simply $2\alpha-1 = 0.365379$ where $\alpha = \Phi(1)-\Phi(-1) = P(|X|<1)$ where $X \sim N(0,1)$. This can be achieved by translating the percentile of $X$ (considering the percen …

This is related to the coupon-collector problem. These random variables have been studied by many people, although I don't recall a particular name for them. See, for example, Anna Pósfai's thesis (ab …

No, you need some other condition if you want to restrict the total variation distance. The total variation distance is maximal between a discrete random variable and a continuous one, and you can tak …

First, here is a counterexample to a simpler statement with $a_0$ fixed at $0$: Let $Y$ be the constant random variable $2$. Let $X$ be $1$ with probability $1/2$, and $3$ with probability $1/2$. Then …

It sounds like you want evenly spaced quantiles of a distribution. If you have to represent a distribution with one number, the median may be a reasonable choice. If $n=9$, then you can use the $10$th …

You can read off the density function as follows: Since a Laplace distribution with mean $0$ is the difference of two IID exponential distributions, the sum of $n$ IID Laplace distributions with mean …

Here is a direct argument. Suppose independent $X_1,X_2 \sim X$, and $X_1+X_2$ is uniform on $[0,1]$. $X$ is supported on $[0,1/2]$. For any $0 \lt \alpha \lt 1/4$, $\alpha = P\left(X_1+X_2 \in [ …

I'll address the second question on the expected value of the sum $K_n$. Let $\phi(x)$ and $\Phi(x)$ be the probability density function and cumulative distribution functions for a standard normal di …

Here is some notation I used in the related problem with some results. Let the probability hat a Brownian motion starting at $0$ returns to home on $[a,b]$ be $h(a,b) = h(b/a)$. By the calculation in …

Getting all-in while behind It is not just when you are ahead that you might want to get all-in against someone who has an information advantage. Suppose the pot is $1$ and the effective stack depth …

Here is an answer to the updated question: Suppose that there are two betting rounds. Darth Vader has three types of hands. Type 1 wins with probability 1. Type 2 is a draw that hits (becomes a winni …

Assume that the process stops when someone can't fit. I believe the distribution of the amount of overshoot is known as a ladder height distribution, and that this is in Feller's classic text, but I …